Finite difference operators with a finite-band spectrum

F. Peherstorfer; P. Yuditskii

doi:10.1007/978-3-7643-8755-6_18

Finite difference operators with a finite-band spectrum

F. Peherstorfer, P. Yuditskii

Johannes Kepler University Linz

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so-called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the 2d+1-diagonal matrix; the d-root of the product of the Green functions (counting their multiplicities) with respect to all infinite points on the surface is the symbol of the shift operator. We demonstrate an application of our general construction to the particular covering, which generate almost periodic CMV matrices recently widely discussed. Then we study an important theme: covering of one spectral surface by another one and the related transformations on the set of multidiagonal operators (so-called Renormalization Equations). We prove several new results dealing with Renormalization Equations for periodic Jacobi matrices (polynomial coverings) and for the case of a rational double covering.

Original language	English
Title of host publication	Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006
Editors	Jan Janas, Pavel Kurasov, Ari Laptev, Ari Laptev, Sergei Naboko, Gunter Stolz
Publisher	Springer International Publishing
Pages	347-389
Number of pages	43
ISBN (Print)	9783764387549
DOIs	https://doi.org/10.1007/978-3-7643-8755-6_18
State	Published - 2009
Externally published	Yes
Event	Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 - Lund, Sweden Duration: 1 Jan 2006 → …

Publication series

Name	Operator Theory: Advances and Applications
Volume	186
ISSN (Print)	0255-0156
ISSN (Electronic)	2296-4878

Conference

Conference	Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006
Country/Territory	Sweden
City	Lund
Period	1/01/06 → …

Bibliographical note

Publisher Copyright:
© 2008 Birkhäuser Verlag Basel/Switzerland.

Funding

Partially supported by the Austrian Founds FWF, project number: P20413-N18 and Marie Curie International Fellowship within the 6th European Community Framework Programme, Contract MIF1-CT-2005-006966.

Funders	Funder number
Austrian Founds FWF	P20413-N18
Sixth Framework Programme	MIF1-CT-2005-006966

Keywords

CMV matrices
Expanding polynomials
Hardy spaces
Iterations
Jacobi matrices
Riemann surfaces
Spectral theory

Access to Document

10.1007/978-3-7643-8755-6_18

Cite this

Peherstorfer, F., & Yuditskii, P. (2009). Finite difference operators with a finite-band spectrum. In J. Janas, P. Kurasov, A. Laptev, A. Laptev, S. Naboko, & G. Stolz (Eds.), Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 (pp. 347-389). (Operator Theory: Advances and Applications; Vol. 186). Springer International Publishing. https://doi.org/10.1007/978-3-7643-8755-6_18

Peherstorfer, F. ; Yuditskii, P. / Finite difference operators with a finite-band spectrum. Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. editor / Jan Janas ; Pavel Kurasov ; Ari Laptev ; Ari Laptev ; Sergei Naboko ; Gunter Stolz. Springer International Publishing, 2009. pp. 347-389 (Operator Theory: Advances and Applications).

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Peherstorfer, F & Yuditskii, P 2009, Finite difference operators with a finite-band spectrum. in J Janas, P Kurasov, A Laptev, A Laptev, S Naboko & G Stolz (eds), Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. Operator Theory: Advances and Applications, vol. 186, Springer International Publishing, pp. 347-389, Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006, Lund, Sweden, 1/01/06. https://doi.org/10.1007/978-3-7643-8755-6_18

Finite difference operators with a finite-band spectrum. / Peherstorfer, F.; Yuditskii, P.
Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. ed. / Jan Janas; Pavel Kurasov; Ari Laptev; Ari Laptev; Sergei Naboko; Gunter Stolz. Springer International Publishing, 2009. p. 347-389 (Operator Theory: Advances and Applications; Vol. 186).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Peherstorfer F, Yuditskii P. Finite difference operators with a finite-band spectrum. In Janas J, Kurasov P, Laptev A, Laptev A, Naboko S, Stolz G, editors, Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. Springer International Publishing. 2009. p. 347-389. (Operator Theory: Advances and Applications). doi: 10.1007/978-3-7643-8755-6_18

Finite difference operators with a finite-band spectrum

Abstract

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Keywords

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